UNIVERSITY OF CALGARY. Solving Multi-objective Optimization Problems in Power Systems. Based on Extended Goal Programming Method.

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1 UNIVERSITY OF CALGARY Solving Multi-objective Optimization Problems in Power Systems Based on Extended Goal Programming Method by Syed Sabbir Ahmed A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING CALGARY, ALBERTA SEPTEMBER, 2013 c Syed Sabbir Ahmed 2013

2 Abstract This thesis proposes an approach to solve multi-objective optimization problems in power systems based on the Extended Goal Programming (EGP) method. In the first part, the EGP method is applied to deterministic multi-objective optimal power flow (MOOPF) problem. The results are compared with classical methods and the efficiency of the EGP method is evaluated. A method for ranking the solutions is introduced to help decision makers choose their preferred solution. In the second part, Taguchi s Orthogonal Array Technique (TOAT) and EGP method are jointly applied to solve probabilistic MOOPF problem with load and renewable generation uncertainties. This approach finds a solution that is robust to uncertain variations in load and renewable power generations. An analysis of the significance of generators ramp rate variation with the degree of robustness of the solution is shown. The results are compared with that of deterministic model and the robustness of the solution is evaluated. i

3 Acknowledgements At first, I would like to take the opportunity to convey my sincere gratitude to Dr. William D. Rosehart for his invaluable guidance and supervision throughout my days in The University of Calgary. Dr. Rosehart, being my mentor, not only guided my research work smoothly, but also helped me to develop myself as a research professional. I feel really very grateful for all of his support that I got here in Calgary. I am equally thankful to my co-supervisor, Dr. Hamidreza Zareipour for his relentless support. I learned so many things from him that my words are not enough to explain. Whenever I faced troubles, I rushed to him and he made things easy for me. I found him as a true guide who guided me in the right path in terms of my research and professional development. He is the first person who introduced me to this great research community. I am also thankful to my peers in the university. Yue, Hamid and Greg helped me always through my bad and good times. Pamela and Vivek, being my first two research mates in Calgary, helped me a lot during my course works. Mahdi, Ali, David, Han and Mo were always there to guide me in the right direction and shared their important research experiences. I am greatly thankful to my Bangladeshi friends in Calgary. Tangim, Subashis, Riad, Shubhrajit, Imran, Ibrahim, Tauhid, Shawn, Hasib, Samiul and Ovy, you guys always gave me a feel that I am in my home in Bangladesh. There are so many loving memories of our friendship, that I cannot forget in my life time. Last but certainly not the least, my family was always there with their huge support from thousands of miles away from Canada. My parents, my younger brother and my grandmother showered me so much love and care that my words are finding no bounds to express Thanks to almighty for gifting me such an awesome family. My research work in Calgary would never be easy without their endless love and care. ii

4 Dedication To my beloved parents, Prof. Rashida Begum and Dr. Enayet Hossain who shaped my life with all the hardest efforts possible iii

5 Table of Contents Abstract i Acknowledgements ii Dedication iii Table of Contents iv List of Tables vii List of Figures viii List of Symbols ix 1 Introduction Overview Literature Review Review of Previous Work on Multi-objective Optimization Problems in Power System Review of Previous Works on Goal Programming Theory Review of Previous Works on Goal Programming to Solve Multi-objective Optimization Problems in Power Systems Review of Previous Work on Solving Multi-objective Optimization Problems in Power System under Uncertainties Summary of Reviews Research Objectives and Motivation Solving multi-objective optimization problems in power systems Solving multi-objective optimization problems in power system under uncertainties Structure of the Thesis Background Review Introduction Basics of Modeling Simple Optimization Problem Single Objective Optimization Problem Multi-objective Optimization Problem Optimal Power Flow Problem Single Objective Optimal Power Flow Problem Multi-objective Optimal Power Flow Problem Probabilistic Single Objective Optimal Power Flow Problem Probabilistic Multi-objective Optimal Power Flow Problem Goal Programming Multiplex Model Lexicographic Goal Programming Model Weighted Goal Programming Model Min-max Goal Programming Model Extended Goal Programming Method Theory of Extended Goal Programming Pareto Efficiency in Multi-objective Optimization Analytic Hierarchy Process iv

6 2.6.1 Taguchi Orthogonal Array Testing (TOAT) Method Value of B Value of M Summary Extended Goal Programming Approach to Solve Multi-objective AC Optimal Power Flow Problem Introduction Methodology Extended Goal Programming Formulation of Multi-objective Optimal Power Flow Selection of Target Goals Achievement and Deviation Level Difference Level Compromise Solution Non-dominated Solutions of Extended Goal Programming Method Ranking Strategy Numerical Results Target Goals Selection of Weights Compromise Between Maximum Achievement and Deviation Level Comparison of Results from Different Models Sensitivity of Weights and Ranking of Optimal Schedules Summary An Approach to Solve Multi-objective Optimal Power Flow Problem under Load and Renewable Generation Uncertainties Introduction Methodology and Modeling Probabilistic MO-OPF Constraints Considering Load and Renewable Generation Uncertainties Converting Probabilistic Constraints into Deterministic Constraints by applying TOAT Robust Multi-objective Optimal Power Flow Formulation Robust Extended Goal Programming Formulation Modified Robust Extended Goal Programming Formulation Solution Methodology Selection of Goals Achievement and Deviation Level Compromise Solution Calculating the Degree of Robustness Numerical Results Selection of Appropriate Orthogonal Array Target Goals Selection of Weights Compromise Between Maximum Achievement and Deviation Level Degree of Robustness v

7 4.3.6 Comparison Between Robust and Non-robust Model Summary Conclusions Bibliography vi

8 List of Tables 2.1 Scale of Relative Importance Generation scenarios for system Z based on Orthogonal Array L 9 (3 4 ) Generation cost, Emission Coefficients, and Real and Reactive Power Limits of Generators for IEEE-30 bus system Best and Worst Values of Each Objective Scale of Relative Importance Comparison of Maximum Achievement and Maximum Deviation for Different Z Values for IEEE-30 Bus System Comparison of Maximum Achievement and Maximum Deviation for Different Z Values for IEEE-118 Bus System Optimal Dispatch Schedule for Different Optimization Models for IEEE-30 Bus System Optimal Dispatch Schedule for Different Optimization Models for IEEE-118 Bus System Comparison of Different Weights for Different Values of Z for IEEE-30 Bus System Comparison of Different Weights for Different Values of Z for IEEE-118 Bus System Best and Worst Values of Each Objective Comparison of Maximum Achievement and Maximum Penalty for Different Z Values for IEEE-14 Bus System Comparison of Maximum Achievement and Maximum Penalty for Different Z Values for IEEE-30 Bus System Degree of Robustness (R deg ) for IEEE-14 bus system robust model Degree of Robustness (R deg ) for IEEE-14 bus system non-robust model Degree of Robustness (R deg ) for IEEE-30 bus system robust model Degree of Robustness (R deg ) for IEEE-30 bus system non-robust model Comparison Between Robust and Non-robust Model for IEEE-14 bus system Comparison Between Robust and Non-robust Model for IEEE-30 bus system 89 vii

9 List of Figures and Illustrations 2.1 Significance of Deviation Variables Pareto Front in a Minimization Problem Pareto Front in a Maximization Problem Hierarchy Tree Pairwise Comparison Matrix for Criterion Compromise Solution in Extended Goal Programming Hierarchy Tree for Group A Pairwise Comparison Matrix for A Difference Level for Different Z values for IEEE-30 Bus System Difference Level for Different Z values for IEEE-118 Bus System Difference Level for Different Optimization Methods for IEEE-30 Bus System Difference Level for Different Optimization Methods for IEEE-118 Bus System Difference Level for Different Set of Weights for IEEE-30 bus Difference Level for Different Set of Weights for IEEE-118 bus Ranking of optimal weight combinations based on based on difference level for IEEE-30 Bus and IEEE-118 Bus Difference Level for Different Set of Weights for IEEE-14 bus Difference Level for Different Set of Weights for IEEE-30 bus R deg for Robust and Non-robust model for IEEE-14 bus R deg for Robust and Non-robust model for IEEE-30 bus Difference Level for Robust and Non-robust model for IEEE-14 Bus and IEEE- 30 bus viii

10 List of Symbols, Abbreviations and Nomenclature Symbol P, P Gj Q, Q Gj a i, b i, c i, d i, e i C(P ) E(P ) P loss α i,β i,γi,ζ i,λ i N B ij G ij P i Q i P Di Q Di V i, V j θ ij P min,p max Q min,q max Vi min,vi max θi min,θi max Definition Real power generation of ith and jth units Reactive power generation of ith and jth units Generation cost coefficients of ith unit Total generation cost in $/h Total emission in tons/h Total transmission loss of system in MW Emission cost coefficients Total number of units Susceptance of line between ith and jth bus Conductance of line between ith and jth bus Real power input at the ith bus Reactive power input at the ith bus Real power demand at ith bus Reactive power demand at ith bus Voltage at ith and jth buses Current and voltage angle between ith and jth bus Minimum and maximum real power generation limits of ith unit Minimum and maximum reactive power generation limits of ith unit Minimum and maximum voltage limits at ith bus Limits of angle between voltage and current at ith bus ix

11 Chapter 1 Introduction 1.1 Overview ven the increasing complexity of power systems, optimization problems in this field, are mostly multi-objective by nature. Power system planning problems [1],[2], reactive power compensation schemes [3], transmission line expansion problems [4], economic-emission load dispatch problems [5], hydrothermal scheduling problems [6], and multi-objective optimal power flow (OPF) problems [7] are well known multi-objective optimization problems in power systems. Many algorithms have been developed for multi-objective optimization problems, including the weighting method [8], the min-max optimum method [9], the weighted minmax method [10], the ɛ-constraint method [8], the utility method [11], the global criterion method [9], and the goal programming method [12]. However, solving multi-objective power systems planning is challenging because of the high complexity of modern power systems with variable loads and intermittent renewable power generators. Thus, developing efficient algorithms to model and solve multi-objective optimization problems in power systems are of interest. The focus of the thesis is to propose an approach to solve multi-objective optimization problems in power systems based on extended goal programming method. In the first part of the thesis, extended goal programming method is applied to a deterministic multi-objective optimal power flow problem.the results are compared with classical goal programming methods and the efficiency of the extended goal programming method is shown. In the second part, Taguchi s Orthogonal Array Technique (TOAT) and extended goal programming method are jointly applied to a probabilistic multi-objective optimal power flow problem with load and renewable generation uncertainties. The aim of this approach is to find a solution which is 1

12 robust to uncertain variations in load and renewable power generations. The results are compared with previously mentioned deterministic problem and the robustness of the solution is shown. 1.2 Literature Review Review of Previous Work on Multi-objective Optimization Problems in Power System Multi-objective optimization problems in power systems are of importance now-a-days. Planning and control for optimal operation of power systems are complex and with increasing adoption of new technologies (e.g. wind power generators, solar panel etc.) in this arena, making it even more complex day by day. As a result, there arises many conflicting issues and objectives in power system optimization to achieve which is difficult to model as classical single objective optimization problem. Therefore, power system optimization problems are essentially modeled as multi-objective optimization problem in current research. Power system planning problem is often considered as a well known optimization problem with multiple objectives such as providing all instantaneous electricity demands, fulfilling every system security criterion[2]. Power system planning problems also includes other multiple objectives such as minimizing long term network operation costs, minimizing total investment cost for transmission and generation processes, minimizing negative environmental impact by electricity generation and transportation, etc[1]. Reactive power compensation schemes are also modeled as multi-objective optimization problem where minimizing transmission system power losses and maximizing active power transfer capacity are two conflicting objectives to optimize[3][7]. From the system operators point of view, it is also reasonable to model the power system optimization problem as a multi-objective optimization problem since there are two conflicting objectives such as minimization of investment and operation cost and maximization of benefits for all agents participating in the market. Power system expansion problem is also considered as a multi-objective optimization problem where the three 2

13 conflicting objectives are such as minimization of network expansion cost, maximization of power system reliability, maximization of effects on energy prices for individual agents [4]. Some other examples of multi-objective optimization problems in power system include environmentally constrained economic dispatch [5], hydrothermal scheduling [6], optimal power flow[7] Review of Previous Works on Goal Programming Theory There are solution methods and algorithms proposed in the literature to solve multi-objective optimization problems. Among them, goal programming[12], [13] is a tool to solve multiobjective optimization problems. It was first proposed by Abraham Charnes and William Cooper in At the beginning stage, goal programming was limited to linear multipleobjective problems. The basic theory of goal programming was extended and modified by Ijiri et al.[14]. With these extension, goal programming was made capable to cope with nonlinear and integer problems and also provided an effective economic interpretation of its result by introducing the concept of multi-dimensional dual. However, there are three major goal programming variants in the literature namely, lexicographic/preemptive,weighted and min-max/chebychev goal programming[15]. In the lexicographic goal programming method, solutions are achieved in a number of priority levels. Each priority level contains a number of unwanted deviations to be minimized. Preferential weights are assigned to design the relative importance of the minimization of the associated deviation variable. If there are any unimportant deviation variables which s minimization is neglected, a preferential weight of zero is assigned. Minimization of deviation variables at a higher priority level is considered more important than that of deviation variables at a lower priority level. This produces a series of sequential optimizations. As the minimal values of the higher priority level optimizations must be maintained, each new optimization has a reduced feasible region than the previous one. Lexicographic goal programming technique is well suited with the problems where decision maker has a pre-defined ordering of the goals in mind and does not want 3

14 any direct trade-off comparisons between goals. On the other hand, direct trade off between all unwanted deviation variables are allowed in weighted goal programming variant. It is also known as non pre-emptive goal programming. Hence weighted goal programming variant gives the decision maker more flexibility than pre-emptive goal programming approach. Chebychev/min-max goal programming, the third major goal programming variant is first proposed by Flavell in 1976[16]. This method uses underlying Chebychev means (L ) of measuring distance. Here, the maximal deviation from any goal is minimized. Therefore, this variant is often regarded as Min-max goal programming. Measuring the Chebychev (L ) distance actually points out the balance between the solutions. In min-max method, the decision maker tries to achieve a balance between achievement of different set of goals where in lexicographic method, some goals are given priorities over others deliberately or in weighted goal programming, where a set of decision variables is chosen, regardless of maintaining any balance, just to make the achievement function lowest [14],[15]. A new goal programming variant is proposed by Carlos Romero in 2001 which is known as extended goal programming[14][17]. The importance and potential of this new method is justified from the utility representation of goal programming. From the utility theory point of view, the weighted and min-max solutions represent two opposite poles. The utility interpretation of weighted goal programming indicates the maximization of a separable and additive utility function in the number of attributes/objectives considered. That also states that weighted goal programming solution provides maximum achievement which means maximum efficiency. On the other hand, the utility interpretation of min-max method indicates the minimization of the maximum deviation from the goal. It states that min-max method provides the a balanced solution between achievements of different goals which means maximum equity. It also shows that the weighted goal programming solution could be extremely biased towards some of the goals while min-max solution could provide poor aggregated performance between different goals [14]. In extended goal programming model, the achievement 4

15 function is formed as such that it encompasses the weighted and min-max goal programming variants in a unified format. Since the weighted and min-max formulation gives solution which resides in two completely different poles, extended goal programming formulation provides a compromise solution between these two opposite concepts of optimizing efficiency and equity which is very much efficient while dealing with conflicting goals [17] Review of Previous Works on Goal Programming to Solve Multi-objective Optimization Problems in Power Systems Goal programming algorithms have been applied to solve multi-objective problems in power system research. Among them, weighted goal programming and min-max goal programming are common and used frequently by the researchers in the literature [18]. In [7], a multi-objective optimal power flow model is proposed to optimize active and reactive power dispatch while maximizing voltage security at the same time. The proposed model is solved using interior point method via goal programming and linearly combined objective functions. In [19], a fuzzy multi-objective mixed integer linear programming model is proposed for secondary voltage control method. Three control objectives have been taken into account: voltage of pilot node should be close enough to the reference value updated by the tertiary voltage control, deviation of the important bus voltage should be least and the reactive power output of the control generators should be adjusted according to the Mvar capabilities. Fuzzy goals are adopted for each objective and ranking of the priority of each objective is proposed based on fuzzy logic method. The entire problem is solved by fuzzy goal programming method. A weighted goal mixed integer programming model for rescheduling of generation power in deregulated markets is proposed in [20]. There are several objectives in the optimization model which includes to achieve the energy and reserve programs based on market bidding, to minimize the total cost and to schedule smooth power changes in the plants. The problem 5

16 is solved using mixed integer goal programming method and the real size application of the model for a Spanish utility has been presented successfully. The computational time is short and hence it is suitable for use in real time by a company in different steps of the sequential process of markets as decision support for obtaining schedules. A fuzzy goal programming formulation of multi-objective optimal power flow is introduced in [21].The membership functions of the defined fuzzy goals are characterized first for measuring the degree of achievement of the aspiration levels of the goals specified in the decision making context. In the solution process, the genetic algorithm is employed to the fuzzy goal programming formulation of the problem for achievement of the highest membership value of the defined membership functions to the extent possible in the decision-making environment. In the GA based solution search process, the conventional Roulette wheel selection scheme [22], arithmetic crossover and random mutation are taken into consideration to reach a satisfactory decision. In [23], a multi-objective problem of optimal planning of distributed generator units in the distribution system is formulated and evaluated using goal programming method along with genetic algorithm. In [24], Genetic Algorithm (GA) based fuzzy goal programming (FGP) technique to multi-objective optimal planning of electric power generation and dispatch problem in power system operation and planning phases is presented. This method is similar to that of [21] but the problem formulation is slightly different since there are three objectives, minimum fuel cost, minimum emission and minimum voltage deviation. The main advantage of the proposed approach is that the use of FGP incorporates the major source of uncertainty in optimal dispatch problem. A multi-objective fuzzy nonlinear goal programming approach is proposed in [25] to minimize wastage of electricity at the source of generation and supply line and maximize sales and profit. The model is implemented in the coal based thermal power plant in India 6

17 in order to generate electricity at a controlled cost which in turn can maximize its sales and profit. Each objective function is optimized separately subject to the constraints of the problem. Optimum values of each objective are calculated. The value of remaining objective functions at each cases are calculated and a pay off matrix is constructed. From the pay off matrix, lower and upper bounds of the objectives are calculated. The membership functions of the maximization of the objective functions and the minimization of the objective functions are constructed and zimmermann model of nonlinear programming is formulated and solved. In [26], a new model based on goal programming is proposed as constant voltage PQ model to solve the optimal reactive power flow in wind generation integrated system. In the CVPQ model, the wind farm bus is considered as PQ node with constant voltage to isolate the wind farms from system influences. Finally the goal programming model of optimal reactive power flow is solved by using a genetic algorithm since it can cope with complex nonlinear goal programming. Artificial intelligence technique is integrated with sensitivity analysis for the formulation and resolution of the optimal reactive power flow problem in [27]. The objectives and constraints are transformed into fuzzy sets and the problem is solved by a fuzzy goal programming algorithm. In [28], a maintenance scheduling problem of thermal generating units under economic and reliability criteria is solved by goal programming model. The problem is formulated as a large scale mixed integer programming problem implemented in the mathematical programming language GAMS and solved by using Optimization Subroutine Library (OSL). A sequential goal programming is used to solve the problem where economic criterion optimization is taken care of in the first place and then the reliability criterion optimization is done. In [29], economic-emission load dispatch (EELD) problem is solved through linear and non-linear goal programming algorithms. For nonlinear goal programming formulation, Box 7

18 complex method is used to minimize the achievement functions. The advantage is the operating limits on the decision variables will be taken care of while generating the feasible points. Equality constraints are transformed into inequality constraints since box complex method cannot handle them. Linear goal programming formulation is solved using sequential or multiphase linear goal priming method. A new algorithm is proposed for nonlinear formulation which overcomes the drawback of selection of step size or starting feasible point. In [30], Fuzzy Goal Programming (FGP) is adopted to handle the multi-objective distributed generator (DG) placement problem incorporating the voltage characteristics of each individual load component. The original objective functions and constraints are transformed into the multi-objective function with fuzzy sets by FGP. The solution of the transformed multi-objective function with fuzzy sets is searched by Genetic Algorithm (GA). Three objectives of Multi-objective Optimal Power Flow (MOPF) problem: cost of generation, system transmission losses, environmental pollution are considered and MOPF problem is attempted sequentially using sequential goal programming (SGP) in [31], [32]. Zangwill transformation has been used for transforming a constrained multi-objective optimal power flow problem into a sequence of unconstrained problems and a standard algorithm of quasi Newton variable metric method has been used for unconstrained minimization. Regret Analysis has been carried out to determine the optimal strategy. The optimal strategy is one for which the regret is minimum. In [33], a goal programming model for the optimal mix and location of renewable energy plants in the north of Spain is proposed. As different types of plants can be placed in each location, the goal is to locate one plant in each place, maximizing the number of plants that are matched with comparable locations, in a way that the total deviations from goals are minimized. The problem was solved by using Lingo. A fuzzy mixed integer goal programming approach for cooking and heating energy planning in rural India is introduced in [34]. The solutions provide energy resource allocations 8

19 at micro level with minimized cost, minimized emission, maximized social acceptance and maximized use of local resources. A fuzzy goal programming model is introduced to develop Oregon s renewable energy portfolio in [35].Portfolio analysis indicates to minimize the costs and maximize the benefits, therefore it has a multi-objective character here. In [36], the energy allocation process is looked at from two points of view: economy and environment. The economic objectives include costs, efficiency, energy conservation, and employment generation. The environmental objectives consider environmental friendliness factors. The objective functions are first quantified and then transformed into mathematical language to obtain a multi-objective allocation model based upon pre-emptive goal programming techniques. The proposed method allows decision-makers to encourage or discourage specific energy resources for the various household end-uses. In [37], a modified extended goal programming model is used with interval programming to model the Economic Emission Load Dispatch (EELD) problem. In this paper, the target goals are considered as interval-valued numbers. The solution is sought then using genetic algorithm. A multi-objective electricity planning problem in Spain is formulated using compromise programming model in [38],. At first, the objective functions are converted into an equivalent compromise programming model. After, the compromise programming model is modified and converted into the extended goal programming model to achieve the set of best-compromise solutions Review of Previous Work on Solving Multi-objective Optimization Problems in Power System under Uncertainties Most of the works discussed are developed to solve deterministic multi-objective optimization problems. Extensions of these approaches have been proposed in the literature to handle the uncertain factors in the optimization problem. There are a number of methods proposed in 9

20 the literature to handle uncertainty in goal programming models. In [39], three approaches are proposed including the uncertain random expected value goal programming, the minmax chance-constrained goal programming and dependent-chance goal programming. In [40], a chance-constrained integer goal programming model for capital budgeting considering the uncertainty in product demand is proposed. In [41], a stochastic chance-constrained goal programming model and algorithm is proposed for oilfield measures. A chance constrained fuzzy goal programming model has been proposed in [42]. A scenario based goal programming model has been proposed in [43] to handle the uncertain factors. In [44], a stochastic goal programming approach based on fuzzy beta is proposed for portfolio selection problem under uncertainty. In [45], a transportation network design problem (NDP) with multiple objectives and demand uncertainty is modeled using stochastic goal programming. A stochastic goal programming model based on mean-variance minimization is proposed in [46]. In [47], a fuzzy goal programming model based on measuring goal attainment value is proposed. A decision support model to help public water agencies allocate surface water among farmers based on stochastic goal programming is proposed in [48].In [49], a location allocation problem under demand and supply uncertainty is modeled based on stochastic chance constrained goal programming theory. An uncertain multi-objective job shop problem is modeled based on fuzzy goal programming in [50]. In [51], a short-term unit commitment problem in a deregulated market environment has been modeled and solved using fuzzy mixed integer goal programming. In this paper, the goals for various objectives are assumed as uncertain due to their characteristics and modeled as fuzzy numbers. Finally, the proposed fuzzy mixed-integer goal programming model is converted into the equivalent crisp model and solved. In [52], a multi-objective optimal power flow problem is modeled based on fuzzy goal programming and then solved using genetic algorithm. Here also the goals are assumed as uncertain and thus modeled as fuzzy numbers. In [53], a power generation and dispatch problem is modeled and solved 10

21 based on fuzzy goal programming theory. In these above research, uncertainties associated with load and renewable generations are not considered. In [54], a single objective DC optimal power flow is solved. Taguchi s Orthogonal Array Technique is used to select the optimal scenarios and finall the problem is solved by interior point method Summary of Reviews Most of the previous works reviewed in earlier sections on solving multi-objective optimization problems, use classical goal programming methods such as weighted goal programming, min-max goal programming etc. However, extended goal programming has not been widely used for solving multi-objective optimization problems in power systems. The scope of this thesis is to implement extended goal programming method to solve multi-objective optimization problems in power systems and show the effectiveness of the method compared to other major goal programming variants. 1.3 Research Objectives and Motivation In the modern power systems with variable loads and uncertain renewable power generators, it is a key issue to obtain an optimal dispatch schedule which can satisfy multiple system objectives (e.g. minimizing generation cost, emission, transmission loss etc.) simultaneously and also robust to the variations in load and renewable power generations. In this thesis, extended goal programming is applied to solve a multi-objective AC optimal power flow problem. In addition, with the integration of wind and solar power in power systems, methods to address the uncertainty of these resources in system operation and planning is becoming important. The later part of this thesis is dedicated to solve a multi-objective AC optimal power flow problem in power systems under uncertainties. 11

22 1.3.1 Solving multi-objective optimization problems in power systems Solving multi-objective optimization problems in power system is important. Multi-objective optimization techniques in power system have number of advantages such as allowing the management of different objectives, simplifying the decision making process by trading off among conflicting objectives, providing information on the consequences of the decision considering all the objectives considered. This thesis considers a multi-objective AC optimal power flow problem to minimize generation cost, emission and transmission power loss simultaneously. Recent study shows that total reported green house gas emissions from 165 Alberta facilities across 15 industrial sectors equaled mega tonnes of carbon dioxide equivalent (Mt CO 2 ) [56]. The consequences of this emission are concerning. Another study reports that coal-fired power generation is likely to cause thousands of early deaths in Alberta and cost the province hundreds of millions of dollars [57] [58]. The total transmission loss in Alberta power systems is estimated 2759 GWhr/ year. This loss costs 240 million CAD annually [59]. Minimizing emission and transmission loss along with generation cost is desired. However, since these objectives are conflicting in nature, it is difficult to find one single solution to the problem. Thus, compromise solution is necessary. Chapter 3 of this thesis proposes an approach to solve multiple conflicting objectives of an AC optimal power flow problem based on extended goal programming theory. Extended goal programming is a method that can provide a compromise solution which is a trade off between maximum achievement of goals and maximum deviation from any of the goal. In this chapter, first a multi-objective AC optimal power flow problem has been transformed to an extended goal programming formulation. After that, target goals for each of the objectives are set using the solution of single objective AC optimal power flow problem for each of the objectives and optimal weights are calculated using analytic hierarchy process. Finally the problem is solved finding a compromise solution and results are compared with other classical goal programming methods. 12

23 A contribution of the work in this chapter is an extended goal programming formulation of multi-objective AC optimal power flow problem. Another contribution is an analysis of significance of the controlling parameter, Z, showing the efficiency of the model comparing to other classical goal programming models. The last contribution of this chapter is proposing a ranking strategy to choose the best routine based on decision makers priorities Solving multi-objective optimization problems in power system under uncertainties A recent report says globally over 27,000 MW of new wind generation capacity was added in 2008 which was 36% more than in 2007 [60]. This growing number of intermittent renewable generator penetration may potentially impact the stability of power system operation.in addition, the impact of uncertain load in the system makes the optimal operation of the system more difficult. Failing to handle these uncertainties, may cause system outage and anticipate huge loss to the government. Thus solving multi-objective optimization problems in power systems under uncertainties is important. Chapter 4 of this thesis considers the deterministic multi-objective optimal power flow problem in Chapter 3 with added uncertainties. This chapter proposes an approach to solve multi-objective optimal power flow problem under load and renewable generation uncertainties. In this chapter, Taguchi s Orthogonal Array Testing technique is used to select a minimum number of testing scenarios with good statistical information in the uncertain space. After selecting the optimal scenarios, the uncertain multi-objective optimal power flow problem is transformed into a robust multi-objective optimization problem. Finally, the problem is modeled based on extended goal programming theory and solved. It is shown that the proposed method can provide a solution that is robust uncertain variations in load and renewable generations and also can satisfy all the conflicting objectives. A contribution of the work in this chapter is modeling a multi-objective AC optimal power flow problem under uncertainties using Taguch s Orthogonal Array Technique and extended goal programming method. The second contribution is an analysis of the significance of 13

24 ramp rate variation with the degree of robustness of the solution. The final contribution of this chapter is a comparison between robust and non-robust solution from the extended goal programming point of view showing the efficiency of the proposed approach. 1.4 Structure of the Thesis Chapter 2: A detail background discussion on modeling single and multi-objective optimization problems from power systems point of view is given in this chapter. Background information is also given on basics of goal programming and its different variants. A discussion on Analytic Hierarchy Process (AHP) is given and detail methodology is shown step by step. Finally, a brief discussion is given on Taguchi s Orthogonal Array Technique (TOAT). Chapter 3: An approach based on extended goal programming to solve multi-objective optimal power flow problem is proposed in this chapter. Analytic Hierarchy Process has been applied to choose optimal weights. A ranking strategy is also presented to choose the best routine based on the decision makers priorities. The model is tested using the standard IEEE-30 and IEEE-118 bus systems. Finally the results are shown and the efficiency of the approach is verified comparing with other classical methods. Chapter 4: In this chapter, an approach to solve multi-objective optimal power flow problem under load and renewable generation uncertainties are presented. Taguchi s Orthogonal Array Technique (TOAT) is used to select optimal scenarios. After that the uncertain multiobjective optimal power flow problem has been transformed into a robust multi-objective optimal power flow problem. Finally the problem is modeled using extended goal programming and solved. The results are shown and the efficiency of the algorithm is verified. Chapter 5: This chapter summarizes the main contributions and conclusions of this thesis. 14

25 Chapter 2 Background Review 2.1 Introduction In this thesis, different methods and techniques are used to model multi-objective optimization problem in the context of power systems, to evaluate the solution quality and to handle the uncertainties in the system for making the model robust. In this chapter, relevant background information is provided on the the basics of modeling different types of optimization problems and optimal power flow problems, basics of goal programming, analytic hierarchy process and Taguchi s Orthogonal Array Technique. 2.2 Basics of Modeling Simple Optimization Problem An optimization problem has been well defined in the literature. An optimization is the problem of finding the best solution out of a set of feasible solution. The solution is called optimal solution for that specific problem. In this section, basic modeling of simple optimization problems are discussed. Optimization problems can be divided into two categories based on the number of objectives. They are single objective and multi-objective. The modeling of these two types of problems is as follows: Single Objective Optimization Problem Single objective optimization problems has one objective to minimize or maximize. Thus the name Single Objective Optimization Problem (SOP) has come. The general form of single objective optimization problem is written as follows: 15

26 max f(x) (2.1) subject to : v j (x) = 0, j = 1, 2, 3,..., p (2.2) g j (x) 0, j = 1, 2, 3,..., p (2.3) which maximizes a real valued function f of x = (x 1, x 2, x 3,..., x n ) subject to sets of constraints g j (x) 0, j = 1, 2, 3,..., p and v j (x) = 0, j = 1, 2, 3,..., p. In this formulation, x is decision vector, and x 1, x 2,..., x n are decision variables. The function f is called the objective function. The set S, S = {x ε R n g j (x) 0, j = 1, 2,..., p} (2.4) is called feasible set. An element x in S is called a feasible solution. A feasible solution x is called the optimal solution of single objective optimization problem if and only if, f(x ) f(x) (2.5) Multi-objective Optimization Problem Single objective optimization problem is related to maximizing or minimizing a single function subject to a number of constraints. However, it has been increasingly recognized that many real-world decision making problems involve multiple, non-commensurable, and conflicting objectives which should be considered simultaneously. As an extension, multiobjective optimization problem is defined as a means of optimizing multiple objective functions subject to a number of constraints, i.e., 16

27 max [f 1 (x), f 2 (x),..., f m (x)] (2.6) subject to : v j (x) = 0, j = 1, 2, 3,..., p (2.7) g j (x) 0, j = 1, 2, 3,..., p (2.8) where f i (x) are objective functions, i = 1, 2,..., m, and g j (x) 0 and v j (x) = 0 are system constraints, j = 1, 2, 3,..., p. When the objectives are in conflict, there is no optimal solution that simultaneously maximizes all the objective functions. For this case, a concept of pareto solution is employed, which means that it is impossible to improve any one objective without sacrificing on one or more of the other objectives. A feasible solution x is said to be a Pareto solution if there is no feasible solution z such that, f i (x) f i (x ), i = 1, 2, 3,..., m (2.9) and f j (x) f j (x) for at least one index j. If the decision maker has a real-valued preference function aggregating the m objective functions, then the aggregating preference function subject to the same set of constraints can be maximized. This model refers to as a compromise model whose solution is called a compromise solution. A well known compromise model is set up by weighting the objective functions, i.e., m max λ i f i (x) (2.10) i=1 subject to : g j (x) 0, j = 1, 2, 3,..., p (2.11) where the weights λ 1, λ 2,..., λ m are non-negative numbers with λ 1 + λ λ m = 1. Note that the solution of (2.8) must be a pareto solution of the original problem. 17

28 2.3 Optimal Power Flow Problem Optimal Power Flow (OPF) is considered one of the most efficient and powerful analyzing tool for the economic operation of power system[61]. Optimal power flow (OPF) in electric power systems is a method of determining the optimal settings of various control variables for minimizing generation cost, voltage deviations, emission cost, transmission losses etc. while satisfying power flow constraints. OPF problem is normally static, non-linear, multiobjective in nature Single Objective Optimal Power Flow Problem A general single objective OPF problem can be expressed as follows, min f(x 1, x 2,..., x n ) (2.12a) St. g m (x 1, x 2,..., x n ) = a (2.12b) b min n x n b max n (2.12c) where f(x 1, x 2,..., x n ) is the function to be minimized, x 1, x 2,..., x n are the control variables associated with the system, g m (x 1, x 2,..., x n ) are the power balance equations of the system, b min n and b max n are the lower and upper limits of the control variables respectively. A sample single objective OPF problem is as follows, min C(P ) = s.t. N i=1 (a i + b i P + c i P 2 ) + di sin(e i (P min P )) (2.13a) P P Di V i V j (G ij cosθij + B ij sinθ ij ) = 0 j i Q Q Di V i V j (G ij sinθij B ij cosθ ij ) = 0 j i V min i V i V max i (2.13b) (2.13c) (2.13d) P min Q min P P max Q Q max (2.13e) (2.13f) 18

29 θi min θ i θi max (2.13g) where P, P Gj are real power generation of ith and jth units, Q, Q Gj are reactive power generation of ith and jth units, a i, b i, c i are generation cost coefficients of ith unit, C(P ) is total generation cost in $/h, N is total number of units, B ij is susceptance of line between ith and jth bus, G ij conductance of line between ith and jth bus, P Di is real power demand at ith bus, Q Di is reactive power demand at ith bus, V i, V j are voltage at ith and jth buses, θ ij are current and voltage angle between ith and jth bus, P min, P max are minimum and maximum real power generation limits of ith unit, Q min, Q max are minimum and maximum reactive power generation limits of ith unit, Vi min, Vi max are minimum and maximum voltage limits at ith bus, θi min, θi max are minimum and maximum limits of angle between voltage and current at ith bus Multi-objective Optimal Power Flow Problem Due to the complex operational perspective of the power system, it is obvious that there are objectives which are to be optimized jointly. Here lies the importance of the multi-objective optimal power flow in which two or more objectives are jointly minimized satisfying the system constraints. A general multi-objective OPF problem can be expressed as follows, min f i (x 1, x 2,..., x n ) i = 1, 2, 3,...p (2.14a) St. g m (x 1, x 2,..., x n ) = a m = 1, 2, 3,...q (2.14b) b min n x n b max n (2.14c) where f i (x 1, x 2,..., x n ) are the objective functions to be minimized, p is the total number of objectives to be minimized, x 1, x 2,..., x n are the control variables associated with the system, g m (x 1, x 2,..., x n ) are the power balance equations of the system, q is the total number of power balance constraints, b min n respectively. and b max n are the lower and upper limits of the control variables 19

30 A sample multi-objective optimal power flow problem is as follows, min C(P ) = N i=1 min E(P ) = 10 2 ( s.t. (a i + b i P + c i P 2 ) + di sin(e i (P min P )) (2.15a) N α i + β i P + γ i P) 2 + ζ i exp(λ i P ) i=1 k=1 (2.15b) N L min P loss = g k [Vi 2 + Vj 2 2V i V j cos(δ i δj)] (2.15c) P P Di V i V j (G ij cosθij + B ij sinθ ij ) = 0 j i Q Q Di V i V j (G ij sinθij B ij cosθ ij ) = 0 j i V min i V i V max i (2.15d) (2.15e) (2.15f) P min Q min θ min i P P max Q Q max θ i θ max i (2.15g) (2.15h) (2.15i) where P, P Gj are real power generation of ith and jth units, Q, Q Gj are reactive power generation of ith and jth units, a i, b i, c i are generation cost coefficients of ith unit, C(P ) is total generation cost in $/h, E(P ) is total emission in tons/h, P loss is total transmission loss of system in MW, Bk ij, Bk 0i, Bk 00 are Kron s loss coefficients, α i, β i, γi, ζ i, λ i are emission cost coefficients, N is total number of units, B ij is susceptance of line between ith and jth bus, G ij conductance of line between ith and jth bus, P Di is real power demand at ith bus, Q Di is reactive power demand at ith bus, V i, V j are voltage at ith and jth buses, θ ij are current and voltage angle between ith and jth bus, P min, P max are minimum and maximum real power generation limits of ith unit, Q min, Q max are minimum and maximum reactive power generation limits of ith unit, Vi min, Vi max are minimum and maximum voltage limits at ith bus, θi min, θi max are minimum and maximum limits of angle between voltage and current at ith bus. 20

31 2.3.3 Probabilistic Single Objective Optimal Power Flow Problem The optimal power flow problems discussed is Section and are deterministic in nature. But in a real-life power system optimization problems, there are a number of uncertain parameters exist. Two of the most common uncertain parameters are load and renewable power generation uncertainties. With the increasing integration of renewable generators into the grid such as wind, solar etc., the optimal power flow problems turn out to be a probabilistic optimization problem. A general probabilistic single objective optimal power flow problem can be expressed as, min f(x 1, x 2,..., x n ) (2.16a) St. g m (x 1, x 2, x 3, x 4..., x n ) = a (2.16b) b min n x n b max n (2.16c) where f(x 1, x 2,..., x n ) is the function to be minimized, x 1, x 2 are the controllable variables associated with the system, x 3, x 4 are the uncertain variables associated with the system, g m (x 1, x 2,..., x n ) are the power balance equations of the system, b min n and upper limits of the control variables respectively. and b max n are the lower A sample probabilistic single objective optimization problem is as follows, min C(P ) = St. N i=1 (a i + b i P + c i P 2 ) + di sin(e i (P min P )) (2.17a) P + P Ri P Di V i V j (G ij cosθij + B ij sinθ ij ) = 0 jɛi Q + Q Ri Q Di V i V j (G ij sinθij B ij cosθ ij ) = 0 jɛi V min i V i V max i (2.17b) (2.17c) (2.17d) P min Q min P P max Q Q max (2.17e) (2.17f) 21

32 θi min θ i θi max (2.17g) where P, P Gj are real power generation of ith and jth units, Q, Q Gj are reactive power generation of ith and jth units, a i, b i, c i are generation cost coefficients of ith unit, C(P ) is total generation cost in $/h, N is total number of units, B ij is susceptance of line between ith and jth bus, G ij conductance of line between ith and jth bus, PDi is uncertain real power demand at ith bus, QDi is uncertain reactive power demand at ith bus, PRi is uncertain renewable real power generation, Q Ri is uncertain renewable reactive power generation, V i, V j are voltage at ith and jth buses, θ ij are current and voltage angle between ith and jth bus, P min, P max are minimum and maximum real power generation limits of ith unit, Q min, Q max are minimum and maximum reactive power generation limits of ith unit, Vi min, Vi max minimum and maximum voltage limits at ith bus, θi min, θi max limits of angle between voltage and current at ith bus. are are minimum and maximum Probabilistic Multi-objective Optimal Power Flow Problem A general probabilistic multi-objective OPF problem can be expressed as follows, min f i (x 1, x 2,..., x n ) i = 1, 2, 3,...p (2.18a) St. g m (x 1, x 2, x 3, x 4..., x n ) = a (2.18b) b min n x n b max n (2.18c) where f i (x 1, x 2,..., x n ) is the function to be minimized, p is the total number of objectives to be minimized, x 1, x 2 are the controllable variables associated with the system, x 3, x 4 are the uncertain variables associated with the system, g m (x 1, x 2,..., x n ) are the power balance equations of the system, b min n respectively. and b max n are the lower and upper limits of the control variables 22

33 A sample probabilistic multi-objective optimal power flow problem is as follows, min C(P ) = min E(P ) = 10 2 ( St. N i=1 (a i + b i P + c i P 2 ) + di sin(e i (P min P )) (2.19a) N α i + β i P + γ i P) 2 + ζ i exp(λ i P ) T ons/hr (2.19b) i=1 N L min P loss = g k [Vi 2 + Vj 2 2V i V j cos(δ i δj)] (2.19c) k=1 P + P Ri P Di V i V j (G ij cosθij + B ij sinθ ij ) = 0 jɛi Q + Q Ri Q Di V i V j (G ij sinθij B ij cosθ ij ) = 0 jɛi V min i V i V max i (2.19d) (2.19e) (2.19f) P min Q min θ min i P P max Q Q max θ i θ max i (2.19g) (2.19h) (2.19i) where P, P Gj are real power generation of ith and jth units, Q, Q Gj are reactive power generation of ith and jth units, a i, b i, c i are generation cost coefficients of ith unit, C(P ) is total generation cost in $/h, E(P ) is total emission in tons/h, P loss is total transmission loss of system in MW, Bk ij, Bk 0i, Bk 00 are Kron s loss coefficients, α i, β i, γi, ζ i, λ i are emission cost coefficients, N is total number of units, B ij is susceptance of line between ith and jth bus, G ij conductance of line between ith and jth bus, P Di is uncertain real power demand at ith bus, Q Di is uncertain reactive power demand at ith bus, P Ri is uncertain renewable real power generation, Q Ri is uncertain renewable reactive power generation, V i, V j are voltage at ith and jth buses, θ ij are current and voltage angle between ith and jth bus, P min, P max are minimum and maximum real power generation limits of ith unit, Q min, Q max and maximum reactive power generation limits of ith unit, Vi min, Vi max maximum voltage limits at ith bus, θi min, θi max are minimum are minimum and are minimum and maximum limits of angle 23

34 between voltage and current at ith bus. 2.4 Goal Programming Goal programming, a methodology for the modeling, solution, and analysis of problems having multiple and conflicting goals and objectives, has often been cited as being the workhorse of multiple objective optimization (i.e., the solution to problems having multiple, conflicting goals and objectives) as based on its extensive list of successful applications in actual practice Multiplex Model Multiplex model is the backbone of goal programming formulation. Any multi-objective optimization problem can be transformed into a multiplex model to fit with goal program method. Suppose a optimization problem is given as follows, max f(x 1, x 2 ) (2.20a) St. g 1 (x 1, x 2 ) a (2.20b) g 2 (x 2 ) b x 0 (2.20c) (2.20d) The multiplex model of the above problem is as follows, min U = (p 1 + n 2 ), f(x 1, x 2 ) (2.21a) St. g 1 (x 1, x 2 ) + η 1 ρ 1 = a (2.21b) g 2 (x 2 ) + η 2 ρ 2 = b (2.21c) x, η, ρ 0 (2.21d) 24

35 Here, a negative deviation variable is added to, and a positive deviation variable is subtracted from, each constraint. In addition,the maximizing objective function is transformed into a minimizing form by simply multiplying the original objective function by a negative one. The new variables (i.e., the negative and positive deviation variables, that have been added to the constraints) indicate that a solution to the problem may result, for a given constraint i, in a negative deviation or a positive deviation or no deviation. That is to say that a goal (be it a hard or soft constraint) can be underachieved, overachieved, or precisely satisfied. In the multiplex formulation, the deviation variables that are to be minimized is appeared in the first (highest priority) term of the achievement function. Once the first term has been minimized, the next term the second term can be dealt with. The algorithm will seek a solution that minimizes the value of this second term, but this must be accomplished without degrading the value already achieved in the higher priority term. A simple multiobjective problem is given below, max Z 1 (x 1, x 2 ) (2.22a) max Z 2 (x 1, x 2 ) (2.22b) St. G 1 (x 1, x 2 ) a (2.22c) G 2 (x 1 ) b G 3 (x 2 ) c x 0 (2.22d) (2.22e) (2.22f) Multiplex form of the above problem is given below, min U = (ρ 1 + ρ 2 + ρ 3 ), (η 4, η 5 ) (2.23a) St. G 1 (x 1, x 2 ) + η 1 ρ 1 = a (2.23b) G 2 (x 1 ) + η 2 ρ 2 = b (2.23c) 25

36 G 3 (x 2 ) + η 3 ρ 3 = c Z 1 (x 1, x 2 ) + η 4 ρ 4 = g 1 Z 2 (x 1, x 2 ) + η 5 ρ 5 = g 2 (2.23d) (2.23e) (2.23f) x, η, ρ 0 (2.23g) To transform an objective into a goal, one must assign some estimate (usually the decision makers preliminary estimate) of the aspired level for that goal. It is assumed here that the aspiration level for Z 1 (x 1, x 2 ) is g 1 units while that of Z 2 (x 1, x 2 ) is g 2 units. In the achievement function in eqn. (2.23a), minimization of negative deviation variables η 1, η 2 and η 3 are ignored since function G 1 (x 1, x 2 ), G 2 (x 1 ), G 3 (x 2 ) must be less than of equal to a, b and c, respectively. Similarly, minimization positive deviation variables ρ 4 and ρ 5 are also ignored since function Z 1 (x 1, x 2 ) and Z 2 (x 1, x 2 ) must be greater than target goals g 1 and g Lexicographic Goal Programming Model In the lexicographic goal programming model, decision makers prioritize their goals into different priority levels such as 1, 2, 3 etc. Each of this priority level may contain one or more goals. If a priority level contains two or more goals, these goals should be weighted as same. The main idea behind this model is that a lower priority level goals must not be achieved at the expense of higher priority goals. That indicates, if the minimum total weighted deviation of priority 1 goals has a value of N 1, then it must be ensured that this value remains same while looking for to minimize the total weighted deviations of priority 2 goals. Similarly, if the weighted deviation of the priority 2 goals are valued as N 2, then this value must remain same while seeking to minimize the total weighted deviation of priority 3 goals. Considering a simple problem, where there are 6 goals such as one goal in priority level 1, two goals in priority level 2 and three goals in priority levels 3. Assuming the detrimental deviations from the goals are listed as, D 1, D 2, D 3, D 4, D 5 and D 6, respectively. Weights are 26

37 assigned as W 1 for priority level 1, W 2, W 3 for priority level 2 and W 4, W 5, W 6 for priority level 3. Based on the concept of lexicographic goal programming model, it will first solve the priority level 1 problem as follows: Subject to Goal Equations Min W 1 D 1 (2.24) F unctional N on negativity Constraints Constraints Assuming the solution yields a minimum objective function value, N 1. Then the following priority level 2 problem is solved as: Min W 2 D 2 + W 3 D 3 (2.25) Subject to Goal Equations F unctional Constraints W 1 D 1 = N 1 N on negativity Constraints Assuming the solution yields a minimum objective function value, N 2. Then the priority level 3 problem is solved as: Min W 3 D 3 + W 4 D 4 + W 5 D 5 (2.26) Subject to Goal Equations F unctional Constraints W 1 D 1 = N 1 27

38 W 2 D 2 + W 3 D 3 = N 2 N on negativity Constraints The solution to the priority level 3 problem is the final optimal solution to the main goal programming problem Weighted Goal Programming Model Weighted Goal Programming is important when decision makers do not have any pre-emptive ordering of the objective functions. Instead of prioritizing the objective functions in different levels, they assign different weights for deviation variable of each objective function in the single priority level. Considering a simple problem, where there are 6 goals, all in a single priority level. Assuming the detrimental deviations from the goals are listed as, D 1, D 2, D 3, D 4, D 5 and D 6, respectively. Weights are assigned as W 1, W 2, W 3, W 4, W 5, W 6, respectively. Based on the concept of weighted goal programming model, the problem is modeled as follows: Min W 1 D 1 + W 2 D 2 + W 3 D 3 + W 4 D 4 + W 5 D 5 + W 6 D 6 (2.27) Subject to Goal Equations F unctional N on negativity Constraints Constraints The solution to this problem is the optimal solution of this weighted goal programming formulation. This solution provides the maximum achievement between the different goals based on the weights assigned to each objective function. Realizing the weighted goal programming, makes sense only if the numerical weights can be assigned to the non-achievement of each goal. If the goals are non-commensurable, weighted goal programming model should not be used since it unifies all the weights in the same achievement function. 28

39 2.4.4 Min-max Goal Programming Model The notion of min-max goal programming method is that the solution sought is the one that minimizes the maximum deviation from any single goal. Considering a simple problem, where there are 3 goals, all in a single priority level. Assuming the detrimental deviations from the goals are listed as, D 1, D 2 and D 3, respectively. Weights are assigned as W 1, W 2 and W 3, respectively. A dummy variable D max is used to measure the maximum deviation from any of the goal. Based on the concept of min-max goal programming, the problem is modeled as follows: Min D max (2.28a) W i D i D max 0, i = 1, 2, 3 (2.28b) Subject to. Goal Equations F unctional N on negativity Constraints Constraints The constraint in eqn. (2.28b) satisfies that deviation W i D i for each objective function i where i = 1, 2, 3, must not be greater than the maximum deviation D max. It is clear that the min-max model, as shown, is simply a single objective optimization problem in which we seek to minimize a single variable. In other words, we seek to minimize the single worst deviation from any one of the goals. It also indicates that min-max goal programming method provides a balanced solution where the maximum deviation from each goal is minimized Extended Goal Programming Method According to the arguments developed earlier sections, Section and 2.4.4, from a preferential point of view the weighted and the Chebyshev goal programming solutions represent two opposite poles. Since the weighted option maximizes the aggregate achievement among 29

40 the goals considered, the results obtained with this option can be biased against the performance achieved by one particular goal [17]. On the other hand, because of the preponderance of just one of the goals, the min-max model can provide results with poor aggregate performance between different goals. The extreme character of both solutions can lead to some cases to possibly unacceptable solutions by the decision maker. A possible modeling solution for this type of problem consists of compromising the maximum achievement of the weighted goal programming model with the maximum deviation from the goal of the min-max model. Thus, the example of the earlier section, Section 2.4.1, can be reformulated with the help of the following multiplex extended goal programming model: min U = (ρ 1 + ρ 2 + ρ 3 ), [(1 Z)δ + Z(η 4 + η 5 )] (2.29a) St. G 1 (x 1, x 2 ) + η 1 ρ 1 = a (2.29b) G 2 (x 1 ) + η 2 ρ 2 = b G 3 (x 2 ) + η 3 ρ 3 = c Z 1 (x 1, x 2 ) + η 4 ρ 4 = g 1 Z 2 (x 1, x 2 ) + η 5 ρ 5 = g 2 (2.29c) (2.29d) (2.29e) (2.29f) (1 Z)η 4 δ 0 (2.29g) (1 Z)η 5 δ 0 (2.29h) δ, x, η, ρ 0 (2.29i) where, the parameter Z weights the importance attached to the minimization of the sum of unwanted deviation variables. For Z = 0, we have a min-max goal programming model. For Z = 1 the result is a weighted goal programming model, and for other values of parameter Z belonging to the interval (0, 1) intermediate solutions between the solutions provided by the two goal programming options are considered. Hence, through variations in the value of parameter Z, compromises between the solution of the maximum aggregate achievement 30

41 and the min-max solution can be obtained. In this sense, this extended formulation allows for a combination of goal programming variants that, in some cases, can reflect a decision makers actual preferences with more accuracy than any single variant Theory of Extended Goal Programming The importance of the extended goal programming model arises from the utility representation of goal programming. From the utility point of view, the weighted and min-max models represent two opposite poles. The weighted method provides maximum achievement of the target goals, which means maximum efficiency. The min-max method provides the most balanced solution: it minimizes the maximum deviation from the goals, which means maximum equity. The weighted solution can be extremely biased toward some of the goals, whereas the min-max solution can provide poor aggregate achievement. These two opposite phenomena are undesirable. The extended goal programming model provides a compromise between these two models. Its achievement function unifies the weighted and min-max models. Thus, it provides a better compromise solution that is efficient for optimization problems with conflicting goals [17]. In this section, a basic multi-objective optimization problem is formed from utility point of view. After that, the formulation of the weighted and the min-max goal programming is shown successively. Finally, the extended goal programming model is formulated combining these two classical models. A basic multi-objective optimization problem is as follows: min f i (x), i = 1, 2, 3,..., q (2.30a) s.t. x F (2.30b) where, f i (x) is the function to minimize, F is the feasible set for variable x, and q is the total number of objectives to optimize. We can define f i (x) as follows: f i (x) + n i p i = t i, i = 1, 2, 3,..., q (2.30c) 31

42 where, n i is the negative deviation variable, p i is the positive deviation variable, and t i is the target value or goal. The deviation variables n i and p i can be expressed as follows [13]: n i = 1 2 [ t i f i (x) + (t i f i (x))], i = 1, 2, 3,..., q (2.30d) p i = 1 2 [ t i f i (x) (t i f i (x))], i = 1, 2, 3,..., q (2.30e) Adding eqn. (2.30d) and (2.30e) gives: n i + p i = t i f i (x), i = 1, 2, 3,..., q (2.30f) Figure 2.1: Significance of Deviation Variables Fig. 2.1(a) and 2.1(b) show that n i and p i are the difference between the optimal solution, f i (x) and the target goal, t i for the maximization and the minimization problem, respectively. Based on the type of the problem, either n i or, p i will be zero. For example, for a maximization problem, p i is always zero. That means, eqn. (2.30f) provides the actual distance from the goal (see fig. 2.1). If the distance is minimized to zero, the optimal solution actually reaches to the goal (see fig. 2.1(c)). Goal programming algorithms basically try to minimize this distance. Eqn. (2.30c) could be written as a utility function [17]: max q i=1 W p i t i f i (x) p, i = 1, 2, 3,..., q (2.31a) 32 s.t. x F (2.31b)

43 Here, W i is the weight attached to the difference between the achievement of the ith goal and its aspiration level, p is a real number in the interval [1, ), and F is the feasible set. Substituting eqn. (2.30f) into eqn. (2.31a), we get: min q i=1 W p i (n i + p i ) p i = 1, 2, 3,..., q (2.32a) s.t. f i (x) + n i pi = t i (2.32b) n i 0 p i 0 (2.32c) x F. (2.32d) This is the utility equivalent of weighted goal programming [17]. The utility equivalent min-max goal programming model can be written as: min [max W i t i f i (x) ], i = 1, 2, 3,..., q (2.33a) s.t. x F (2.33b) A more clear interpretation of the optimization model described from eqn. (2.33a)-(2.33b) is as follows: min δ (2.34a) s.t. W i (n i + p i ) δ, i = 1, 2, 3,..., q (2.34b) f i (x) + n i p i = t i (2.34c) n i 0 p i 0 (2.34d) x F (2.34e) where, δ is the maximum deviation from any goal. By combining the weighted goal programming model described from eqn. (2.32a)-(2.32d) and the min-max goal programming model described from eqn. (2.34a)-(2.34e), we can form the extended goal programming model: min (1 Z)δ + Z q (α i n i + β i p i ) p (2.35a) i=1 33

44 i = 1, 2, 3,..., q s.t. (1 Z)(α i n i + βip i ) δ (2.35b) f i (x) + n i p i = t i (2.35c) n i 0 p i 0 (2.35d) x F (2.35e) where, α i and β i are the weights attached to the negative and positive deviation variables, respectively and parameter Z weights the importance attached to the minimization of the weighted sum of unwanted deviation variables. Parameter p indicates the importance of maximum deviation, δ compared to deviation variables. For balanced comparison, the value of p is normally chosen as 1. Z = 0 gives the min-max formulation, and Z = 1 gives the weighted formulation. Values of Z between 0 and 1 give a compromise between these two formulations. 2.5 Pareto Efficiency in Multi-objective Optimization In multi-objective optimization, it is difficult to find a single solution which satisfies all the objectives. Rather, it provides a set of candidate solutions. For comparison among the candidate solutions, Pareto dominance and Pareto optimality are widely used. A solution is in the Pareto set if there is no single solution exists that improves at least one of the objectives without degrading any other objectives. Mathematically, a decision vector x = [x 1, x 2, x 3, x 4,...x n ] is considered to Paretodominate the decision vector y = [y 1, y 2, y 3, y 4,...y n ], in a minimization problem, if and only if: iɛ {1, 2, 3,...N}, f i (x) f i (y), jɛ {1, 2, 3,...N} : f j (x) < f j (y) (2.36a) (2.36b) 34

45 Pareto dominance is used to compare and rank decision vectors. x dominates y indicates that f i (x) is better than f i (y) for all i. A solution m is considered as Pareto optimal if and only if there does not exist another solution that dominates it. It also means that solution m can not be improved in one of t he objectives without affecting at least one of the objectives. The corresponding vector f(m) is called the Pareto dominant vector or non-dominated vector. The set of all Pareto optimal solutions is called Pareto optimal set. The corresponding objective vectors are said to be on the Pareto front. Figures 2.2 and 2.3 illustrate two examples of Pareto front in case a biobjective minimization and maximization problem. Figure 2.2: Pareto Front in a Minimization Problem Pareto-optimal sets can be obtained by different multi-objective approaches and other methods. After that, the decision-maker has to select one unique solution from these sets for system implementation. However, selecting a single solution from the Pareto optimal set is difficult, specially when the number of objectives are more than two. Therefore, meaningful research has been carried out to support the decision maker during this post-pareto analysis 35

46 Figure 2.3: Pareto Front in a Maximization Problem phase[62]. Also, solution methods that provide necessary information on the trade offs between different objective functions are of importance. 2.6 Analytic Hierarchy Process The analytic hierarchy process (AHP) proposed by Saaty [63] is widely used for multi-criteria decision support. In the AHP method, a complex decision-making problem is modeled as a hierarchical structure of goals, primary criteria, alternatives, and subcriteria. It uses the decision maker s pairwise comparison to provide the order in which the factors affect a decision, the consistency of the pairwise comparison matrix, and finally a prioritized list of the decisions to be taken. The main steps of the AHP method [64] are briefly discussed below: 1) Step 1: Break down the problem into a hierarchical structure including goals, primary criteria, subcriteria, and alternatives. The overall goal is placed at the top and then primary 36

47 criteria, subcriteria, and the set of alternatives are placed in the hierarchy levels. The detail of this step is discussed below: The first step in the analytic hierarchy process is to model the problem as a hierarchy. An AHP hierarchy is a structured means of modeling the decision at hand. It consists of an overall goal, a group of options or alternatives for reaching the goal, and a group of factors or criteria that relate the alternatives to the goal. The criteria can be further broken down into subcriteria, sub-subcriteria, and so on, in as many levels as the problem requires. A criterion may not apply uniformly. In that case the criterion is divided into subcriteria indicating different intensities of the criterion and these intensities are prioritized through comparisons under the parent criterion. To better understand AHP hierarchies, consider a decision problem with a goal to be reached, three alternative ways of reaching the goal, and four criteria against which the alternatives need to be measured. Such a hierarchy can be visualized as a diagram like the one immediately below, with the goal at the top, the three alternatives at the bottom, and the four criteria in between. There are useful terms for describing the parts of such diagrams: Each box is called a node. A node that is connected to one or more nodes in a level below it is called a parent node. The nodes to which it is so connected are called its children. Applying these definitions to the figure 2.4 below, the goal is the parent of the four criteria, and the three criteria are children of the goal. Each criterion is a parent of the two Alternatives. Note that there are only two Alternatives, but in the figure, each of them is repeated under each of its parents. 2) Step 2: Collect input from the decision makers and form the pairwise comparison matrix. The matrix is formed for each element with respect to the level immediately above in the hierarchy. Thus, primary criteria are evaluated based on their importance to the goal, subcriteria are evaluated based on their importance to the primary criteria, and alternatives are evaluated based on their importance to their parent subcriteria. The detail of this step 37

48 Figure 2.4: Hierarchy Tree is discussed below: AHP uses decision maker s pairwise comparison. For quantifying pairwise comparison, a scale of related importance should be used. By using this scale, decision makers can easily provide their preferences. A standard scale of relative importance is shown in Table 2.1 below: Table 2.1: Scale of Relative Importance Intensity Definition 1 Equally important 3 Somewhat more important 5 Much more important 7 Very much more important 9 Absolutely more important 2,4,6,8 Intermediate values The next, is to forming pairwise comparison matrix for each element in the hierarchy tree. The matrix is formed for each of the element in the hierarchy tree with respect to the level immediately above. For example, in Figure 2.4, Criterion 1, 2 and 3 should be evaluated based on their importance to the Goal. Similarly Alternatives should be evaluated based on their importance to the Criterion 1, 2 and 3. A sample pairwise comparison matrix criterion 1 is shown in figure 2.5 below: This figure shows the pairwise comparison between Alternative 1 and 2. 5/1 in the second element of the first row indicates that Alternative 2 in 5 times more important than 38

49 Figure 2.5: Pairwise Comparison Matrix for Criterion1 Alternative 1. 1/5 in the first element of the second row indicates the vice versa. 3) Step 3: Find the maximal eigenvalue and the associated eigenvector [65], [66] for each matrix to get the relative weights of the primary criteria, sub criteria, and alternatives. 4) Step 4: Aggregate the relative weights of the primary criteria and sub criteria to obtain a composite priority for each criterion and each level. This process gives an overall priority vector for all the alternatives that helps the decision maker to select the best option Taguchi Orthogonal Array Testing (TOAT) Method TOAT is a method to select minimum number of testing scenarios with good statistical information in the uncertain space. It has been proven that TOAT is able to select optimal representative testing scenarios from the possible combinations in additive and quadratic models. Compared with Monte Carlo simulation, the number of testing scenarios of TOAT are much less, therefore, the computational burden is also less. Additionally, achieving scenarios with TOAT is much more simple than other scenario reduction methods. Below is a tutorial where the methodology of applying TOAT is discussed: Assume a system z, depicted by z = Z(x 1, x 2,..., x K, ũ 1, ũ 2,..., ũ M ) where x 1, x 2,..., x K are controllable factors and ũ 1, ũ 2,..., ũ M are uncertain factors. To make the system ro- 39

50 bust, the uncertain factors ũ 1, ũ 2,..., ũ M are considered by a series of scenarios. If the total number of uncertain factors are very big, it is not feasible to consider all the scenarios all together. Rather, some representative scenarios are selected to ease the computational burden. For the simplicity of the problem, for each uncertain variable ũ 1, ũ 2,..., ũ M, a total of B representative levels are selected. Thus, the total number of operating states of ũ 1, ũ 2,..., ũ M will be B M. But B M could be a very big number if M is large. To handle this difficulty, Taguchi s orthogonal array technique (TOAT) is used. In this technique, scenarios are selected by orthogonal arrays. An orthogonal array matrix can be denoted by L H (B M ) where H and M are the number of rows and columns respectively and B is the number of matrix element levels. An OA L 9 (3 4 ) can be shown as follows: L 9 (B 4 ) = Based on the system z, an appropriate OA for that particular system can be obtained from OA libraries[74]. OA is selected based on the following considerations Value of B The number of element levels 1, 2,..., B in OA matrix indicated the number of representative levels of uncertain factors in the system. Based on Taguchi s theory, if an uncertain factor ũ M has a linear effect on the system z, then ũ M should have two testing levels. For symmetrically 40

51 distributed ũ M, then µ(ũ M ) σ(ũ M ) and µ(ũ M ) + σ(ũ M ) should be chosen. If an uncertain factor ũ M has a quadratic effect on the system z, then ũ M should have three testing levels. For symmetrically distributed ũ M, µ(ũ M ) 3/2σ(ũ M ), µ(ũ M ) and µ(ũ M ) + 3/2σ(ũ M ) should be chosen[54] Value of M Value of M is directly related to the number of uncertain factors of the system. If the number of uncertain factors in the system is M, then the number of columns in OA is also chosen as M. In this problem, the value of M is chosen based on the number of uncertain loads and generations. After selecting the appropriate values of B and M, the OA is formed for the system Z. The rows of OA indicates the number of scenarios selected. For system Z, determined by OA L H (B M ), a total of H scenarios are formed and H is much smaller than B M. For example, assume that there are four uncertain variables ũ 1, ũ 2,...ũ 4 in the system and two levels are selected for each uncertain variable. Determined by the number of variables and the number of variable levels, OA L 9 (3 4 ) is selected to form the testing scenarios. The way of forming nine testing scenarios according to L 9 (3 4 ) is shown in Table 2.2. In this case, the total of nine testing scenarios are formed, which is less than the number of full combinations 3 4. Therefore, by applying TOAT, the number of testing scenarios are minimized. Table 2.2: Generation scenarios for system Z based on Orthogonal Array L 9 (3 4 ) No. testing scenarios Variable levels ũ 1 ũ 2 ũ 3 ũ 4 1 u 1 (1) u 2 (1) u 3 (1) u 4 (1) 2 u 1 (1) u 2 (2) u 3 (2) u 4 (2) 3 u 1 (1) u 2 (3) u 3 (3) u 4 (3) 4 u 1 (2) u 2 (1) u 3 (2) u 4 (3) 5 u 1 (2) u 2 (2) u 3 (3) u 4 (1) 6 u 1 (2) u 2 (3) u 3 (1) u 4 (2) 7 u 1 (3) u 2 (1) u 3 (3) u 4 (2) 8 u 1 (3) u 2 (2) u 3 (1) u 4 (3) 9 u 1 (3) u 2 (3) u 3 (2) u 4 (1) 41

52 The following features of an OA ensure that TOAT achieves representative testing scenarios which are uniformly distributed over the uncertain operating space. 1) In each OA column, every level occurs H/B times. For example, in L 9 (3 4 ), 1,2 and 3 occur H/B = 9/3 = 3 times. 2) In any two columns, the level combinations appear the same number of times. In L 9 (3 4 ), 1 1, 1 2, 1 3, 2 1, 2 2, 2 3, 3 1, 3 2, 3 3 occur once in any two columns. 2.7 Summary In this chapter, background information needed for this research is presented. A general overview of forming simple optimization problem is provided. Optimal power flow problem is discussed in detail from classical formulation to the probabilistic multi-objective formulation. A detailed discussion on different variants of goal programming is given. Difference between various methods and importance of using extended goal programming is shown. The use of analytical hierarchy process for choosing optimal weight combination is discussed in detail. Finally, taguchi method of orthogonal arrays is discussed. 42

53 Chapter 3 Extended Goal Programming Approach to Solve Multi-objective AC Optimal Power Flow Problem 3.1 Introduction In this chapter, an extended goal programming approach to solve multi-objective AC optimal power flow problem is proposed. A ranking strategy is also presented to choose the best routine based on the decision makers priorities. The model is tested using the standard IEEE-30 and IEEE-118 bus systems. Goal programming is an efficient tool for modeling multi-objective optimization problems. However, extended goal programming has not been widely used in modeling multi-objective optimization problems in power systems. Most of the works use classical goal programming methods such as weighted goal programming, min-max goal programming, fuzzy goal programming etc. Extended goal programming is a relatively new variant of goal programming where the achievement function is a convex combination of weighted and min-max goal programming. Thus, it can provide a better compromise solution than the classical goal programming models. In this chapter, a multi-objective AC optimal power flow problem has been modeled and solved based on extended goal programming method to achieve a compromise solution. Extended goal programming is chosen because it can provide an efficient compromise solution which is a trade off between two opposite characteristics of the model, maximum achievement of goal and maximum deviation from goal. Unlike [37], [38], a multi-objective AC optimal power flow problem has been formulated directly into the extended goal programming model. Also, a different strategy for analytic hierarchy process (AHP) to select weight 43

54 considering different groups of decision makers is presented and significance of Z parameter in the extended goal programming model is demonstrated. The first contribution of the work in this chapter is an extended goal programming formulation of multi-objective AC optimal power flow problem.the second contribution is an analysis of significance of the Z parameter, showing the efficiency of the model comparing to other classical goal programming models. The last contribution of this chapter is proposing a ranking strategy to choose the best routine based on decision makers priorities. The rest of this chapter is organized as follows: Section 3.2 describes the methodology and modeling used in this chapter. Section 3.3 describes the numerical results obtained from different IEEE case study models. Finally, Section 3.4 provides summary and conclusion. 3.2 Methodology In this section, the methodology and the modeling of the proposed approach are discussed in detail. The considered multi-objective AC optimal power flow has three objectives to minimize. They are generation cost, emission and transmission power loss. This problem is a deterministic optimization problem since the uncertainties in the power systems such as load variation, renewable generation variation is not considered. An extended goal programming formulation is formulated and finally the solution is achieved step by step. The methodology and modeling can be summarized as follows: Step 1) First a multi-objective AC optimal power flow problem is formulated considering three objectives, namely generation cost, emission and transmission loss, respectively. Step 2) Extended goal programming theory has been applied. Then, the considered traditional multi-objective AC optimal power flow problem has been transformed to an extended goal programming formulation. Step 3) Target goals for each of the objectives are calculated. Analytic hierarchy process has been employed to calculated the optimal weights for each of the objective function based 44

55 on decision makers choice. Then the extended goal programming model is solved for different values of Z. Step 4) Achievement and deviation level for each of the solution for different values of Z is calculated and the compromise solution, which is a trade off between the maximum achievement and deviation level, is selected. Comparison with other classical goal programming model is made and hence, it shown that extended goal programming formulation has advantages over these models. Step 5) Finally the model is solved for different weight combinations, calculated by analytic hierarchy process and a ranking is proposed to choose the best compromise solution based on decision makers choice Extended Goal Programming Formulation of Multi-objective Optimal Power Flow In this section, an extended goal programming formulation of muti-objective optimal power flow problem has been proposed. The model can be written as below, N i=1 min (1 Z)δ + Z(w 1 p 1 + w 2 p 2 + w 3 p 3 ) p (3.1a) St. (1 Z)(u i n i + w i p i ) δ (3.1b) (a i + b i P + c i P 2 ) + di sin(e i (P min P )) + n1 p 1 = t 1 (3.1c) N 10 2 (α i + β i P + γ i P 2 + d i exp(e i P )) + n 2 p 2 = t 2 (3.1d) i=1 N L k=1 g k [V 2 i + V 2 j 2V i V j cos(δ i δj)] + n 3 p 3 = t 3 (3.1e) P P Di V i V j (G ij cosθij + B ij sinθ ij ) = 0 jɛi Q Q Di V i V j (G ij sinθij B ij cosθ ij ) = 0 jɛi V min i V i V max i (3.1f) (3.1g) (3.1h) P min Q min P P max Q Q max (3.1i) (3.1j) 45

56 θi min θ i θi max (3.1k) where, Z is the controlling parameter, w 1,w 2 and w 3 are the weights attached to each positive deviation variable p 1, p 2 and p 3, respectively, u i is the weight attached to the ith negative variable n i, δ is the maximum deviation from the goal,p, P Gj are real power generations of ith and jth units, Q, Q Gj are reactive power generations of ith and jth units, a i, b i, c i are generation cost coefficients of ith unit, Bk ij, Bk 0i, Bk 00 are Kron s loss coefficients, α i, β i, γi, ζ i,λ i are emission cost coefficients, N is total number of units, B ij is susceptance of line between ith and jth bus, G ij is conductance of line between ith and jth bus, P Di is real power demand at ith bus, Q Di is reactive power demand at ith bus, V i, V j are voltages at ith and jth buses, θ ij is current and voltage angle between ith and jth bus, P min, P max are minimum and maximum real power generation limits of ith unit Q min, Q max and maximum reactive power generation limits of ith unit, Vi min, Vi max maximum voltage limits at ith bus, θi min, θi max between voltage and current at ith bus. are minimum are minimum and are minimum and maximum limits of angle Selection of Target Goals One objective at a time is considered. First, the minimum generation cost is considered. To calculate the generation cost goal, a single objective OPF problem is formulated. The solution of this problem gives the minimum generation cost. The emission and transmission loss for the minimum generation cost are also found. In the same way, the minimum emission and the minimum transmission loss are calculated. These minimum values are lower bounds on the solution of the multi-objective optimization problem, so they are selected as the target goals i.e. t best i. The worst-case values of these three objectives are recorded, and these give the worst possible value of the objective functions i.e. t worst i. t best i is the best possible value for the ith objective and t worst i is the worst possible value for the ith objective. 46

57 3.2.3 Achievement and Deviation Level The three objectives are expressed in different functions and units, so the optimal values need to be normalized for a fair comparison. A well-known method is used to normalize the achievement level for each objective [67]. The achievement of an objective is t achieve i = 1 tbest i t best i t optimum i t worst i = toptimum i t best i t worst i t worst i (3.2a) where t achieve i is the normalized achievement level of the ith objective,, and t optimum i actual optimized value for the ith objective. is the The value of t achieve i achievement level is 1. When t optimum i is bounded between 0 and 1. When t optimum i = t best i, the normalized = t worst i, the normalized achievement level is 0. When an objective has an achievement level of 1, it fully satisfies the desired goal. When the level is 0, it completely fails to satisfy the goal. The normalized deviation value of the ith objective is t deviation i = 1 t achieve i. (3.2b) The value of t deviation i is also bounded between 0 and 1. A value of 0 indicates that the objective achieves the goal and there is no deviation. A value of 1 indicates that it completely fails to achieve the goal Difference Level To compare the solution quality based on these two phenomena, maximum achievement and maximum deviation, an unique measure is taken. It is defined as difference level. The difference level provides the absolute difference between maximum deviation and maximum achievement. The difference level of a solution is calculated as follows: L diff = t achieve max t deviation (3.3) max 47

58 For an ideal solution, maximum achievement, t achieve max should be 1 and maximum deviation, t deviation max should be 0. Thus, the ideal value for difference level of a solution L diff should be 1. Similarly, the worst value for difference level of a solution L diff should be 0. Thus, difference level is an unified approach to measure the solution quality based on maximum achievement and maximum deviation of/from any of the goals Compromise Solution The extended goal programming formulation is a convex combination of weighted and minmax goal programming [14], [17]. The extended goal programming model of the multiobjective OPF problem is solved for different values of Z bounded between 0 and 1. The maximum achievement and deviation levels are then calculated for a particular value of Z. The weighted model provides a maximum achievement level by worsening the maximum deviation from the goals for some objectives. The min-max model provides a more balanced solution by minimizing the maximum deviation from the goals while lowering the maximum achievement level. In the extended formulation intermediate values of Z can provide a set of compromise solutions to the decision makers. Extended goal programming hence finds a compromise between maximum achievement and minimized maximum deviation. This characteristic can be explained from the Figure 3.1. The main target for a multiobjective optimization problem is to find a solution which satisfies the target goals. Hence, the maximum achievement level of 1 and maximum deviation level of 0 are desirable. But for the conflicting nature of different objectives, it is actually unachievable. Thus, solutions that provide a high maximum achievement level and low maximum deviation are chosen. In the figure, P, Q are R denote min-max, weighted and extended goal programming solutions, respectively. It is seen that P D < QE < RF and AR > BQ > P C. It explains that min-max solution at P is providing the lowest maximum deviation and weighted solution at R is providing the highest maximum achievement which is good. However, min-max solution at P has the lowest maximum achievement level and weighted solution at R has the 48

59 Figure 3.1: Compromise Solution in Extended Goal Programming highest maximum deviation. Hence, these solutions are not desirable. On the other hand, extended solution at Q provides a lower maximum deviation than weighted solution and higher maximum achievement than min-max solution. Thus, extended goal programming solution provides a compromise between weighted and min-max solutions Non-dominated Solutions of Extended Goal Programming Method Goal Programming solutions can provide Pareto dominant solution which is in-efficient. Research has been carried out to overcome this draw back. In [68], the methods and techniques to achieve non-dominated solutions from goal programming has been discussed. It is suggested in this paper that goal programming method can provide non-dominated solutions if the target goals are set too conservatively. Thus, selecting the target goals strictly is necessary. In our research, we set the target goals conservatively by solving single optimal power flow problem for each of the objectives. Hence, each of the target goals are the solu- 49

60 tion to the single objective optimal power flow and no better solution can be achieved for multi-objective problem beyond this. Because the other objectives are considered as relaxed during selecting the target goals. Although a extended goal programming can find all the Pareto optimal solutions by changing weights of the objective function for a multi-objective optimization problem, we are interested in one unique solution that is the best compromise depending upon the weights assigned. Unlike, classical multi-objective optimization methods, extended goal programming method finds a single solution which is a trade off between maximum achievement level and maximum deviation level. For classical multi-objective approach, it is often very difficult to find a single solution from the Pareto curve. Hence, extended goal programming is useful for finding a single solution compromising different conflicting objectives and considering the particular weights assigned Ranking Strategy A ranking system is used to find the best solution based on two parameters, maximum achievement and maximum deviation. The decision maker may be uncertain about the best set of weights. The AHP method [63] is used to select different sets of weights and calculate the maximum achievement and maximum deviation. Based on these two parameters, rankings for different values for Z are obtained. These help the decision makers to choose the best solution given their preferences. 3.3 Numerical Results The multi-objective extended goal programming model is solved for IEEE-30 bus and IEEE- 118 bus systems. GAMS and a PC with an Intel Core i3 processor of 2.53 GHz and 6 GB of Ram are used. The IEEE-30 bus data is used from Table 3.1 and the other data is acquired from [69]. The IEEE-118 bus data are acquired from [69],[70]. The cost and 50

61 emission coefficients of the generators for the IEEE-118 bus system are taken from [71] Target Goals Table 3.2 lists the target goals and worst-case values for the IEEE-30 bus and the IEEE- 118 bus. The target goals are obtained by solving a single-objective optimization for each objective. Table 3.1: Generation cost, Emission Coefficients, and Real and Reactive Power Limits of Generators for IEEE-30 bus system Generator G1 G2 G3 G4 G5 G6 Real Power Limit (MW) PG min PG max Reactive Power Limit (MVar) Q min G Q max G Generation Cost Coefficient a b c d e Emission Cost Coefficient α β γ ζ λ Table 3.2: Best and Worst Values of Each Objective t best i IEEE-30 Bus t worst i t best i IEEE-118 Bus t worst i Generation Cost $/h $/h $/h $/h Emission Cost kg/h kg/h kg/h 5510 kg/h Transmission Loss 1.72 MW 3.58 MW MW MW Selection of Weights To show the efficiency of the extended goal programming, weights can be chosen arbitrarily. However, to address the multi-objective problem from system operator s point of view ana- 51

62 lytic hierarchy process (AHP) has been used. Analytic hierarchy process is widely used for decision making in various research field [63], [64], [65], [66]. The AHP is used to select the weights. For the multi-objective OPF problem, the decision makers can be divided into four major groups: 1) Group A: Emphasize generation cost objective 2) Group B: Emphasize emission cost objective 3) Group C: Emphasize power loss objective 4) Group D: Emphasize all three objectives Table 3.3: Scale of Relative Importance Intensity Definition 1 Equally important 3 Somewhat more important 5 Much more important 7 Very much more important 9 Absolutely more important 2,4,6,8 Intermediate values To focus the contribution and analyze the significance of the extended goal programing method, any of the groups can be considered. In this case, Group A is selected randomly. For the pairwise comparison, a standard scale [63] as defined in Table 3.3 is used. Depending on the types of the decision makers, different weight combinations can be calculated; three possibilities are considered, each with a different emphasis on the generation cost. They are: 1) Set A 1 : Extreme emphasis on generation cost, very low emphasis on other objectives 2) Set A 2 : Strong emphasis on generation cost, moderate emphasis on emission cost, and very low emphasis on power loss 3) Set A 3 : Moderate emphasis on generation cost, low emphasis on emission cost, and very low emphasis on power loss The hierarchy tree is then formed as shown in Fig Since the same decision-maker group is providing three different sets of weights, it can be assume that they are equally 52

63 Set of Weights Set A1 (1) Generation Cost (9) Emission Cost (1) Set A2 (1) Generation Cost (7) Emission Cost (5) Set A3 (1) Generation Cost (5) Emission Cost (3) Level 1 Level 2 Power Loss (1) Power Loss (1) Power Loss (1) Figure 3.2: Hierarchy Tree for Group A important. The value is set to 1 for each element in level 1. For the three sets, based on the definition given above, the value for each objective is selected from Table 3.3, as shown in Fig Then three 3 3 pairwise comparison matrices are formed for the three sets. The principal eigenvector for each matrix is calculated [65] based on Step 3 discussed in Section 2.5 in Chapter 2. For example, Fig. 3.3 shows the pairwise comparison matrix for Set A 1. Each element of this matrix shows the relative importance of one objective over another. For example, the value 9/1 at the 2nd element of the 1st row means, the generation cost is 9 times more important than emission cost in this case. Finally, the vector of priorities based on Step 4 discussed in Section 2.5 in Chapter 2 is calculated [72]. In this case, it is P = [1.91, 0.81, 0.28]. After normalization, the vector of priorities is P norm = [0.64, 0.27, 0.09]. This indicates the relative weights for the generation cost, emission cost, and power loss. 53

64 Gen. Cost Emission Power Loss Gen. Cost 1 9/1 9/1 Emission 1/9 1 1/1 Power Loss 1/9 1/1 1 Figure 3.3: Pairwise Comparison Matrix for A Compromise Between Maximum Achievement and Deviation Level Table 3.4 shows the results for the IEEE-30 bus system. For Z = 1 the extended formulation is a classical weighted model. The achievement level for each objective is calculated from eqn (3.2a), and the deviation level is calculated from eqn (3.2b). When Z = 1, the maximum achievement is 0.99 and the maximum deviation is This model provides the highest achievement but also the highest deviation. For Z = 0 the extended formulation is a classical min-max model. When Z = 0, the maximum achievement is 0.94 and the maximum deviation is This model gives a balanced solution by minimizing the maximum deviation, but the maximum achievement is lower. Although the change is small, it might not be acceptable to a decision maker who emphasizes the achievement of the goal. Also, the goals are measured in dollars per hour, so a small reduction in the achievement could add a large penalty. For example, an increase of 0.01 in the achievement level of the gen. cost objective for the IEEE-118 bus system saves $/hr. That means, it can save $/day and $/yr. To find a compromise solution, intermediate values of Z such as 0.3, 0.5, and 0.8 are chosen. For Z = 0.5, the maximum achievement is 0.97, and the maximum deviation is 0.67, which is better than the worst-case value of It shows that 54

65 by compromising only a factor of 0.02 in the achievement level, we can improve the deviation level by Practically, here we can reduce the emission by 62.4 kg/day by compensating only 0.72 MW/day. Hence, this intermediate value provides a better compromise solution. Finally, the efficiency of the extended goal programming solution is shown by calculating the difference level of the solution based on maximum achievement and maximum deviation. The difference level is calculated from eqn. (3.3). It is found that for Z = 0.5, the difference level is calculated as 0.30 which is the highest among the other values of Z considered. For an ideal solution, the difference level should be 1. Since extended goal programming solution is providing the maximum difference level, it is considered as a better compromise solution. Difference level for different Z values for IEEE-30 bus are illustrated in Figure 3.4. Table 3.4: Comparison of Maximum Achievement and Maximum Deviation for Different Z Values for IEEE-30 Bus System Z Cost Generation ($/hr) Emission (Kg/hr) Loss (MW ) Achievement Level (1-0) Generation Emission Loss Maximum Deviation Level (1-0) Generation Emission Loss Maximum Difference Level (1-0) A similar phenomenon is observed for the IEEE-118 bus system in Table 3.5. For Z = 1 the extended formulation is a classical weighted model. When Z = 1, the maximum achievement is 0.94 and the maximum deviation is This model provides the highest achievement but also the highest deviation. For Z = 0 the extended formulation is a classical min-max model. When Z = 0, the maximum achievement is 0.87 and the maximum deviation is This model gives a balanced solution by minimizing the maximum deviation, but the maximum achievement is lower. For Z = 0.5, the maximum achievement is 0.93, and the maximum deviation is 0.34, which is better than the worst-case value of It 55

66 Figure 3.4: Difference Level for Different Z values for IEEE-30 Bus System shows that by compromising only a factor of 0.01 in the achievement level, we can improve the deviation level by Practically, here we can reduce the emission by kg/day by compensating only MW/day. Hence, this intermediate value provides a better compromise solution. The efficiency of the extended goal programming solution is shown by calculating the difference level of the solution based on maximum achievement and maximum deviation. It is found that for Z = 0.5, the difference level is calculated as 0.59 which is the highest among the other values of Z considered. For an ideal solution, the difference level should be 1. Since extended goal programming solution is providing the maximum difference level, it is considered as a better compromise solution. Difference level for different Z values for IEEE-118 bus are illustrated in figure Comparison of Results from Different Models To analyze the performance of our model, we must carefully check the value of the maximum achievement, maximum deviation and difference level. The desired values are 1, 0 and 1, 56

67 Table 3.5: Comparison of Maximum Achievement and Maximum Deviation for Different Z Values for IEEE-118 Bus System Z Cost Generation ($/hr) Emission (Kg/hr) Loss (MW ) Achievement Level (1-0) Generation Emission Loss Maximum Deviation Level (1-0) Generation Emission Loss Maximum Difference Level (1-0) Figure 3.5: Difference Level for Different Z values for IEEE-118 Bus System 57

68 respectively. For the IEEE-30 bus system, Table 3.6 compares different single and multiobjective models. The single-objective models give the highest maximum achievement levels. However, the maximum deviation levels are 0.78, 1, and 1, respectively. This indicates that one or more objectives totally deviate from the goal. Thus, these simple models are not efficient; the goal programming models give better results. For the weighted method, the maximum achievement is 0.99 and the maximum deviation is For the min-max method, these values are 0.84 and 0.64, respectively: the maximum deviation is improved, but the maximum achievement is lower. In the extended model, the maximum achievement is 0.97, which is very close to that of the weighted solution, and the maximum deviation is only Thus, the method provides a compromise between the weighted and min-max models. Table 3.6: Optimal Dispatch Schedule for Different Optimization Models for IEEE-30 Bus System Single Multi-Objective Gen Co OPF Emission OPF Loss OPF Weighted Min-max Extended Gen. Cost ($/h) Emission (kg/h) Loss (MW) Max. Achievement Max. Deviation Difference Level This characteristic of the solution can be better observed from the difference level values achieved by different methods. Single objective emission OPF and loss OPF provides a difference level of 0 which indicates the inefficiency of these solutions. Among the other methods, extended goal programming solution provides the highest value for difference level, 0.30, that verifies the efficiency of the solution. Difference level for different optimization methods for IEEE-30 bus are illustrated in figure 3.6. Similarly for the IEEE-118 bus system, Table 3.7 compares different single and multiobjective models. The single-objective models give the highest maximum achievement levels. However, the maximum deviation levels are 1, 1, and 0.69, respectively. This indicates that 58

69 Figure 3.6: Difference Level for Different Optimization Methods for IEEE-30 Bus System Table 3.7: Optimal Dispatch Schedule for Different Optimization Models for IEEE-118 Bus System Single Multi-Objective Gen Co OPF Emission OPF Loss OPF Weighted Min-max Extended Gen. Cost ($/h) Emission (kg/h) Loss (MW) Max. Achievement Max. Deviation Difference Level